Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(25,26)(27,36)(28,33)(29,34)(30,32)(31,37)(35,42)(38,49)(39,43)(40,47)(41,46)(44,50)(45,51)(48,53)(52,54)(55,56);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,38)(29,37)(30,39)(32,45)(33,44)(34,36)(35,43)(40,50)(41,52)(42,51)(46,55)(47,49)(48,54)(53,56);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(27,37)(30,42)(31,43)(33,46)(34,35)(36,45)(38,50)(41,53)(44,54)(47,48)(49,55);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,38)(29,40)(30,41)(31,44)(32,46)(34,47)(35,48)(36,49)(37,50)(39,52)(42,53)(43,54)(45,55)(51,56);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(27,37)(28,40)(31,36)(33,47)(39,51)(41,53)(43,45)(46,48)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437803);